We denote the set of all Borel probability measures on \(X\)  by \(\operatorname{\mathcal {M}}(X)\)  and equip it with the weak\(^*\)  topology, which in our setting is known to be Polish
[1]}.
Recall that \(\mu _n\)  converges to \(\mu \)  in the weak\(^*\)  topology
if and only if for every \(f \in C_b(X)\) 
(i.e., \(f\)  is a bounded, continuous,
real-valued function on \(X\) ) we have \(\int f\, d\mu _n \rightarrow \int f\, d\mu \) .
If \(X\)  is compact, then \(\operatorname{\mathcal {M}}(X)\)  is also compact [1]}.
Although our main results are of a topological nature, it is convenient for us to fix once
and for all a particular metric \(D\)  for \(\operatorname{\mathcal {M}}(X)\) . We choose the Prohorov metric associated to our fixed metric \(\rho \)  on \(X\)  and given by
\(D(\mu ,\nu )=\inf \lbrace \varepsilon >0: \mu (A)\le \nu (A^\varepsilon )+\varepsilon \text{ for all Borel sets }A\subset X\rbrace , \) 
